Causation or Correlation?
In her book Football Management (2010), Sue Bridgewater reminded us of the idea that a soccer manager is never more than six games away from the sack. When a manager falls through a bottom level of average points per game, this has been labelled the ‘trapdoor’.
Similarly, she identified a ‘honeymoon’ period for the incoming manager, who gains uplift after the shock of the change in the six or so games thereafter. Analysing these trapdoor and honeymoon periods for Premier League managers during the period 1992 to 2008, Bridgewater identified a number of plausible causes for the improvement, including players keen to impress the new manager and an improved playing culture instigated by a new charismatic manager.
Yet, by games 12 to 18, she also found that the points benefit of changing the manager had largely vanished, suggesting that there is only a short-lived gain and that that managerial changes in the Premier League do little to improve performance in the long run.
Regression to the mean
Analysing managerial turnover across 18 seasons (1986 to 2004) in the Eredivisie, Bas Ter Weel revealed noticeable patterns of prior decline and subsequent improvement centred on the sacking of one manager and the appointment of a new one. Crucially, however, almost the same pattern could be observed where managers had not been sacked. How so? Ter Weel was unequivocal in his explanation:
“If managers do not matter for differences in performance... and quality does not vary across managers, the only observed performance change following turnover would be mean reversion.”
Mean reversion, also known as regression to the mean, has been widely discussed in the context of Leicester City’s and Chelsea’s performances during the 2015/16 season. The phenomenon was first uncovered by Sir Francis Galton, the Victorian polymath, as he experimented with the heredity of sweet peas.
In cross breeding trials, Galton noted a tendency for the size of the offspring to show a smaller distribution than that of the parents. Crucially, whilst the offspring of larger parents tended to be smaller, the offspring of smaller parents tended to be larger.
Galton described this tendency as reversion or regression to the mean. It is important to realise that there is no requirement for any purpose for this regression in a deterministic sense; nothing is causing it. Rather, it merely represents a random (causeless) process that sees extremes become less extreme. David Sally, co-author of The Numbers Game: Why Everything You Know About Football is Wrong, emphasises the point:
“In the same way that water seeks its own level, numbers and series of numbers will move towards the average, move towards the ordinary. The extraordinary... is followed by the ordinary... the ordinary is what happens. The average is what happens more often than not.”
In other words, sacked mangers are sacked not because they are bad, but because they have been unlucky. By the same token, the performance of a new manager gives the appearance of improvement simply because it is less bad than the extraordinary that preceded it, and closer to the average.
Ter Weel does caution against the idea that managers have no influence whatsoever on the quality of a team. Rather, he sees a selection at play, where the best managers end up at the best clubs, and the worst managers at the worst clubs. Of course, when one manager is sacked, the pulling power of the club will likely be largely restricted to managers of similar calibre and with similar financial demands. Ter Weel’s research has been replicated for other soccer leagues, most particularly in Bundesliga and Serie A.
Implications for bettors
Bettors looking to exploit managerial sackings will need to be aware of mean reversion when assessing the win expectations of a club. The betting odds of a team with a new manager may very well reflect the myth that he brings with him an improved likelihood of winning. But if this represents little more than a random process driven by changes in luck, those odds might very well be inaccurate. Those who understand reversion to the mean should be better able to exploit those errors made by other bettors believing in a cause that is, to all intents and purposes, illusory.